Some geometrical facts

Let's summarize some of the geometrical properties we have observed in the last section. Let's start with parallel and orthogonal straight lines:

Theorem 1

Consider two straight lines $g$ and $h$ with slope $a$ and $b$ .

$g$ and $g$ are parallel, if they have the same slope: $g \parallel h \iff a=b$
Two lines are orthogonal (that is, they for a right angle), if the product of the two slopes is $-1$ : $g \perp h \iff ab=-1$ (instead of the condition $ab=-1$ can we also use $a=-\frac{1}{b}$ or $b=-\frac{1}{a}$ ).

Exercise 1

Consider two linear function $f$ and $g$ . Are they parallel, orthogonal, or neither?

$f(x)=-3x+2$ and $g(x)=6-3x$
$f(x)=10x+1$ and $g(x)=-0.1x-1$
$f(x)=4x+3$ and $g(x)=0.25x+1$

Solution

parallel, because same slope $-3$
orthogonal, because $10\cdot (-0.1)= -1$
neither, because the have not the same slope, and their product is not $-1$ .

We also have looked at the intercept theorem, which comes in handy from time to time. Let us state it here in more detail:

Theorem 2: Intercept theorem

Consider two lines that intersect at point $P$ , and two lines $g$ and $h$ that traverse these two lines, resulting in the intersection points $A$ , $B$ , $A^\prime$ and $B^\prime$ (see figure below). The following is then true:

g \parallel h \iff \frac{PA^\prime}{PA}=\frac{PB^\prime}{PB}=\frac{A^\prime B^\prime}{AB}

Here, with $AB$ we mean the distance between $A$ and $B$ , and so on.

Exercise 2

Use the intercept theorem to determine the length $x$ in the figure below.

Solution

$\frac{10}{4}=\frac{3+x}{3}\rightarrow 7.5=3+x\rightarrow x=\underline{4.5}$ .